I would say that an idiom is a set phrase compared to speech (speaking) or writing that is said to be idiomatic or not.
English, like all languages, is filled with idioms as defined by the OP:
“- having a meaning that cannot be derived from the conjoined meanings of its elements (such as ride herd on for "supervise").”
However, speech and writing is said to be idiomatic or not, if it reflects natural speech in a language. A text or speech can be said to be idiomatic or not. That refers to whether or not it sounds natural or not to a native speaker of the language in question, all other factors being equal.
Here is one paper on idiomaticity, or an explanation of what is natural sounding in a language. Typically, one says a text or speech is idiomatic or unidiomatic. idiomaticity
Whether a text or a person's speech is idiomatic or not can only be determined by speakers of a language. So, if you want to judge a person's speech, you are the authority. This sounds crazy I know. But it really isn't. The translation and interpretation professions call for judging speech and text as being idiomatic or unidiomatic as a matter of course. It's the bane of the profession.
A text or speech may or may not be idiomatic at many levels: use of words, use of tenses, collocations, incomplete or incorrect idioms, general flow, overall register (not mixing slang, colloquial and other types of expression incorrectly in a speech act) etc.
Idiomatic usage is of particular concern to translators and interpreters since they strive to make anything they are translating or interpreting sound natural in the target language.
As a recent example, one can examine the interpretation of Putin's speech by his interpreter: speech- spoken language and idiomaticity
I won't post everything, but here are some red flags re the interpreter's speech, which at times is not entirely idiomatic (not natural sounding in English).
PUTIN: If I may, |I throw in some two cents|. We talked to |Mr.
President|, including this subject as well. We are aware of the stance
of President Trump. I think that we |as major oil and gas power| and I
think the United States is a major gas and oil power as well [...]
|Nor we are interested| in driving prices up, because it will |drain live juices from all other sectors of the economy|
An idiom mistake: to give one's two cents, not “throw in one's two cents” (regarding a topic).
Non-idiomatic usage: “nor we are interested”; “drain live juices from sectors of the economy”; missing determiner.
“Live juices” is not an idiom in English (It actually made me laugh). Not sure if this was the literal translation from Russian, or, attempting to say something like: to drain power (the juice) away from certain sector of the economy.
Juice in English can mean electrical power but it is not used in the plural except in the idiom: get one's juices flowing, which is rather slangy. But here, it really stands out as non-natural when one is listening to the text as a native English speaker.
Verb tense coupled with idiom mistake: “If I may, I will give” or “I am going to give my two cents.”
Speaking about having the ball in our court in Syria. President Trump has just mentioned that we have successfully concluded the world |football| cup. Speaking of |the football|, actually, Mr. President, I will give this ball to you and now the ball is in your court.
- An idiom: correct: now the ball is in your court.
Idiomatic issues: speaking of football, not “the football”. Also, if one wants to be picky, there is a mixed metaphor here: football and tennis. I cannot judge if this existed in Putin's speech or if the interpreter fumbled the ball.
The interpreter needed to say: I will hit the ball to you (tennis) or I will kick the ball to you. And the ball is now yours.
Please note: Mr. Putin's counterpart is not an eloquent speaker at all. And I feel sorry for the person who had to interpret him into Russian, as often the features of Trump's speech are barely literate. However, it is idiomatic though often idiotic.
- Idioms are set phrases, such as to ride herd on; (put or have) the ball in someone's court;
- Idiomatic text or speech is natural speech to a speaker of the language. The linguistic term is idiomaticity, but it's usually expressed in every day language as: idiomatic or non-idiomatic. A text or speech (a person speaking) might actually be completely idiomatic (natural sounding) and not even contain any idioms. Conversely, speech can be filled with idioms and if expressed incorrectly, a person's speech will sound unidiomatic.
Idiomaticity issues in text or speech can range from minor mistakes in using idioms to all kinds of other usage issues such as incorrect verb tenses, vocabulary usage, word collocations, absence of determiners, incorrect deictic usage (this or that, for example), etc. One can, determine, whether a person's speech or writing is idiomatic overall in a particular language. Interpreters and translators deal with these problems everyday. Their holy grail is always striving to be idiomatic in their translation or interpretation work.
(Note: this does not apply necessarily to literary texts, which are a bit different).
Challenge to readers who are not translators or interpreters, who therefore might not really understand the issues at stake:
How do you know to say in English "Thank God that was an easy test." rather than "Thanks God that was an easy test."? Hmm?